The appearance of stripes near the membrane in cross-flow filtration of dyed solutes in water is modeled as a hydrodynamic stability problem of parallel shear flow in a channel of small width.
Spanwise harmonic perturbations in thickness of the concentration boundary layer give spanwise variations in permeate flux which cause cells of streamwise vorticity that are sustained by the shear flow. States of neutral stability are determined by linear theory, giving the spacing of stripes as function of a modified Reynolds number based on permeate flux, channel width, and its ratio to the boundary layer thickness. The present simple model predicts instability to occur at considerably higher values of modified Reynolds number than those prevailing in experiments showing stripes.
- Concentration polarization
- Hydronamic stability